<!doctype html>
<html>
<head>
<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>

<link href='https://fonts.googleapis.com/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text/css' /><style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color: #ffffff; --text-color: #333333; --select-text-bg-color: #B5D6FC; --select-text-font-color: auto; --monospace: "Lucida Console",Consolas,"Courier",monospace; --title-bar-height: 20px; }
.mac-os-11 { --title-bar-height: 28px; }
html { font-size: 14px; background-color: var(--bg-color); color: var(--text-color); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; -webkit-font-smoothing: antialiased; }
h1, h2, h3, h4, h5 { white-space: pre-wrap; }
body { margin: 0px; padding: 0px; height: auto; inset: 0px; font-size: 1rem; line-height: 1.42857; overflow-x: hidden; background: inherit; }
iframe { margin: auto; }
a.url { word-break: break-all; }
a:active, a:hover { outline: 0px; }
.in-text-selection, ::selection { text-shadow: none; background: var(--select-text-bg-color); color: var(--select-text-font-color); }
#write { margin: 0px auto; height: auto; width: inherit; word-break: normal; overflow-wrap: break-word; position: relative; white-space: normal; overflow-x: visible; padding-top: 36px; }
#write.first-line-indent p { text-indent: 2em; }
#write.first-line-indent li p, #write.first-line-indent p * { text-indent: 0px; }
#write.first-line-indent li { margin-left: 2em; }
.for-image #write { padding-left: 8px; padding-right: 8px; }
body.typora-export { padding-left: 30px; padding-right: 30px; }
.typora-export .footnote-line, .typora-export li, .typora-export p { white-space: pre-wrap; }
.typora-export .task-list-item input { pointer-events: none; }
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  #write { padding-left: 20px; padding-right: 20px; }
}
#write li > figure:last-child { margin-bottom: 0.5rem; }
#write ol, #write ul { position: relative; }
img { max-width: 100%; vertical-align: middle; image-orientation: from-image; }
button, input, select, textarea { color: inherit; font: inherit; }
input[type="checkbox"], input[type="radio"] { line-height: normal; padding: 0px; }
*, ::after, ::before { box-sizing: border-box; }
#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p, #write pre { width: inherit; }
#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p { position: relative; }
p { line-height: inherit; }
h1, h2, h3, h4, h5, h6 { break-after: avoid-page; break-inside: avoid; orphans: 4; }
p { orphans: 4; }
h1 { font-size: 2rem; }
h2 { font-size: 1.8rem; }
h3 { font-size: 1.6rem; }
h4 { font-size: 1.4rem; }
h5 { font-size: 1.2rem; }
h6 { font-size: 1rem; }
.md-math-block, .md-rawblock, h1, h2, h3, h4, h5, h6, p { margin-top: 1rem; margin-bottom: 1rem; }
.hidden { display: none; }
.md-blockmeta { color: rgb(204, 204, 204); font-weight: 700; font-style: italic; }
a { cursor: pointer; }
sup.md-footnote { padding: 2px 4px; background-color: rgba(238, 238, 238, 0.7); color: rgb(85, 85, 85); border-radius: 4px; cursor: pointer; }
sup.md-footnote a, sup.md-footnote a:hover { color: inherit; text-transform: inherit; text-decoration: inherit; }
#write input[type="checkbox"] { cursor: pointer; width: inherit; height: inherit; }
figure { overflow-x: auto; margin: 1.2em 0px; max-width: calc(100% + 16px); padding: 0px; }
figure > table { margin: 0px; }
thead, tr { break-inside: avoid; break-after: auto; }
thead { display: table-header-group; }
table { border-collapse: collapse; border-spacing: 0px; width: 100%; overflow: auto; break-inside: auto; text-align: left; }
table.md-table td { min-width: 32px; }
.CodeMirror-gutters { border-right: 0px; background-color: inherit; }
.CodeMirror-linenumber { user-select: none; }
.CodeMirror { text-align: left; }
.CodeMirror-placeholder { opacity: 0.3; }
.CodeMirror pre { padding: 0px 4px; }
.CodeMirror-lines { padding: 0px; }
div.hr:focus { cursor: none; }
#write pre { white-space: pre-wrap; }
#write.fences-no-line-wrapping pre { white-space: pre; }
#write pre.ty-contain-cm { white-space: normal; }
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.md-fences-adv-panel { width: 100%; margin-top: 10px; text-align: center; padding-top: 0px; padding-bottom: 8px; overflow-x: auto; }
#write .md-fences.mock-cm { white-space: pre-wrap; }
.md-fences.md-fences-with-lineno { padding-left: 0px; }
#write.fences-no-line-wrapping .md-fences.mock-cm { white-space: pre; overflow-x: auto; }
.md-fences.mock-cm.md-fences-with-lineno { padding-left: 8px; }
.CodeMirror-line, twitterwidget { break-inside: avoid; }
svg { break-inside: avoid; }
.footnotes { opacity: 0.8; font-size: 0.9rem; margin-top: 1em; margin-bottom: 1em; }
.footnotes + .footnotes { margin-top: 0px; }
.md-reset { margin: 0px; padding: 0px; border: 0px; outline: 0px; vertical-align: top; background: 0px 0px; text-decoration: none; text-shadow: none; float: none; position: static; width: auto; height: auto; white-space: nowrap; cursor: inherit; -webkit-tap-highlight-color: transparent; line-height: normal; font-weight: 400; text-align: left; box-sizing: content-box; direction: ltr; }
li div { padding-top: 0px; }
blockquote { margin: 1rem 0px; }
li .mathjax-block, li p { margin: 0.5rem 0px; }
li blockquote { margin: 1rem 0px; }
li { margin: 0px; position: relative; }
blockquote > :last-child { margin-bottom: 0px; }
blockquote > :first-child, li > :first-child { margin-top: 0px; }
.footnotes-area { color: rgb(136, 136, 136); margin-top: 0.714rem; padding-bottom: 0.143rem; white-space: normal; }
#write .footnote-line { white-space: pre-wrap; }
@media print {
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  .typora-export-show-outline .typora-export-sidebar { display: none; }
  figure { overflow-x: visible; }
}
.footnote-line { margin-top: 0.714em; font-size: 0.7em; }
a img, img a { cursor: pointer; }
pre.md-meta-block { font-size: 0.8rem; min-height: 0.8rem; white-space: pre-wrap; background: rgb(204, 204, 204); display: block; overflow-x: hidden; }
p > .md-image:only-child:not(.md-img-error) img, p > img:only-child { display: block; margin: auto; }
#write.first-line-indent p > .md-image:only-child:not(.md-img-error) img { left: -2em; position: relative; }
p > .md-image:only-child { display: inline-block; width: 100%; }
#write .MathJax_Display { margin: 0.8em 0px 0px; }
.md-math-block { width: 100%; }
.md-math-block:not(:empty)::after { display: none; }
.MathJax_ref { fill: currentcolor; }
[contenteditable="true"]:active, [contenteditable="true"]:focus, [contenteditable="false"]:active, [contenteditable="false"]:focus { outline: 0px; box-shadow: none; }
.md-task-list-item { position: relative; list-style-type: none; }
.task-list-item.md-task-list-item { padding-left: 0px; }
.md-task-list-item > input { position: absolute; top: 0px; left: 0px; margin-left: -1.2em; margin-top: calc(1em - 10px); border: none; }
.math { font-size: 1rem; }
.md-toc { min-height: 3.58rem; position: relative; font-size: 0.9rem; border-radius: 10px; }
.md-toc-content { position: relative; margin-left: 0px; }
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.md-toc-h2 .md-toc-inner { margin-left: 2em; }
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.md-toc-h4 .md-toc-inner { margin-left: 6em; }
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.md-toc-h6 .md-toc-inner { margin-left: 10em; }
@media screen and (max-width: 48em) {
  .md-toc-h3 .md-toc-inner { margin-left: 3.5em; }
  .md-toc-h4 .md-toc-inner { margin-left: 5em; }
  .md-toc-h5 .md-toc-inner { margin-left: 6.5em; }
  .md-toc-h6 .md-toc-inner { margin-left: 8em; }
}
a.md-toc-inner { font-size: inherit; font-style: inherit; font-weight: inherit; line-height: inherit; }
.footnote-line a:not(.reversefootnote) { color: inherit; }
.reversefootnote { font-family: ui-monospace, sans-serif; }
.md-attr { display: none; }
.md-fn-count::after { content: "."; }
code, pre, samp, tt { font-family: var(--monospace); }
kbd { margin: 0px 0.1em; padding: 0.1em 0.6em; font-size: 0.8em; color: rgb(36, 39, 41); background: rgb(255, 255, 255); border: 1px solid rgb(173, 179, 185); border-radius: 3px; box-shadow: rgba(12, 13, 14, 0.2) 0px 1px 0px, rgb(255, 255, 255) 0px 0px 0px 2px inset; white-space: nowrap; vertical-align: middle; }
.md-comment { color: rgb(162, 127, 3); opacity: 0.6; font-family: var(--monospace); }
code { text-align: left; vertical-align: initial; }
a.md-print-anchor { white-space: pre !important; border-width: initial !important; border-style: none !important; border-color: initial !important; display: inline-block !important; position: absolute !important; width: 1px !important; right: 0px !important; outline: 0px !important; background: 0px 0px !important; text-decoration: initial !important; text-shadow: initial !important; }
.os-windows.monocolor-emoji .md-emoji { font-family: "Segoe UI Symbol", sans-serif; }
.md-diagram-panel > svg { max-width: 100%; }
[lang="flow"] svg, [lang="mermaid"] svg { max-width: 100%; height: auto; }
[lang="mermaid"] .node text { font-size: 1rem; }
table tr th { border-bottom: 0px; }
video { max-width: 100%; display: block; margin: 0px auto; }
iframe { max-width: 100%; width: 100%; border: none; }
.highlight td, .highlight tr { border: 0px; }
mark { background: rgb(255, 255, 0); color: rgb(0, 0, 0); }
.md-html-inline .md-plain, .md-html-inline strong, mark .md-inline-math, mark strong { color: inherit; }
.md-expand mark .md-meta { opacity: 0.3 !important; }
mark .md-meta { color: rgb(0, 0, 0); }
@media print {
  .typora-export h1, .typora-export h2, .typora-export h3, .typora-export h4, .typora-export h5, .typora-export h6 { break-inside: avoid; }
}
.md-diagram-panel .messageText { stroke: none !important; }
.md-diagram-panel .start-state { fill: var(--node-fill); }
.md-diagram-panel .edgeLabel rect { opacity: 1 !important; }
.md-fences.md-fences-math { font-size: 1em; }
.md-fences-advanced:not(.md-focus) { padding: 0px; white-space: nowrap; border: 0px; }
.md-fences-advanced:not(.md-focus) { background: inherit; }
.typora-export-show-outline .typora-export-content { max-width: 1440px; margin: auto; display: flex; flex-direction: row; }
.typora-export-sidebar { width: 300px; font-size: 0.8rem; margin-top: 80px; margin-right: 18px; }
.typora-export-show-outline #write { --webkit-flex: 2; flex: 2 1 0%; }
.typora-export-sidebar .outline-content { position: fixed; top: 0px; max-height: 100%; overflow: hidden auto; padding-bottom: 30px; padding-top: 60px; width: 300px; }
@media screen and (max-width: 1024px) {
  .typora-export-sidebar, .typora-export-sidebar .outline-content { width: 240px; }
}
@media screen and (max-width: 800px) {
  .typora-export-sidebar { display: none; }
}
.outline-content li, .outline-content ul { margin-left: 0px; margin-right: 0px; padding-left: 0px; padding-right: 0px; list-style: none; overflow-wrap: anywhere; }
.outline-content ul { margin-top: 0px; margin-bottom: 0px; }
.outline-content strong { font-weight: 400; }
.outline-expander { width: 1rem; height: 1.42857rem; position: relative; display: table-cell; vertical-align: middle; cursor: pointer; padding-left: 4px; }
.outline-expander::before { content: ""; position: relative; font-family: Ionicons; display: inline-block; font-size: 8px; vertical-align: middle; }
.outline-item { padding-top: 3px; padding-bottom: 3px; cursor: pointer; }
.outline-expander:hover::before { content: ""; }
.outline-h1 > .outline-item { padding-left: 0px; }
.outline-h2 > .outline-item { padding-left: 1em; }
.outline-h3 > .outline-item { padding-left: 2em; }
.outline-h4 > .outline-item { padding-left: 3em; }
.outline-h5 > .outline-item { padding-left: 4em; }
.outline-h6 > .outline-item { padding-left: 5em; }
.outline-label { cursor: pointer; display: table-cell; vertical-align: middle; text-decoration: none; color: inherit; }
.outline-label:hover { text-decoration: underline; }
.outline-item:hover { border-color: rgb(245, 245, 245); background-color: var(--item-hover-bg-color); }
.outline-item:hover { margin-left: -28px; margin-right: -28px; border-left: 28px solid transparent; border-right: 28px solid transparent; }
.outline-item-single .outline-expander::before, .outline-item-single .outline-expander:hover::before { display: none; }
.outline-item-open > .outline-item > .outline-expander::before { content: ""; }
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.md-inline-math-container mjx-container { zoom: 0.95; }
mjx-container { break-inside: avoid; }
.md-alert.md-alert-note { border-left-color: rgb(9, 105, 218); }
.md-alert.md-alert-important { border-left-color: rgb(130, 80, 223); }
.md-alert.md-alert-warning { border-left-color: rgb(154, 103, 0); }
.md-alert.md-alert-tip { border-left-color: rgb(31, 136, 61); }
.md-alert.md-alert-caution { border-left-color: rgb(207, 34, 46); }
.md-alert { padding: 0px 1em; margin-bottom: 16px; color: inherit; border-left: 0.25em solid rgb(0, 0, 0); }
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.md-alert-text-important { color: rgb(130, 80, 223); }
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:root {
    --side-bar-bg-color: #fafafa;
    --control-text-color: #777;
}

@include-when-export url(https://fonts.googleapis.com/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext);

/* open-sans-regular - latin-ext_latin */
  /* open-sans-italic - latin-ext_latin */
    /* open-sans-700 - latin-ext_latin */
    /* open-sans-700italic - latin-ext_latin */
  html {
    font-size: 16px;
    -webkit-font-smoothing: antialiased;
}

body {
    font-family: "Open Sans","Clear Sans", "Helvetica Neue", Helvetica, Arial, 'Segoe UI Emoji', sans-serif;
    color: rgb(51, 51, 51);
    line-height: 1.6;
}

#write {
    max-width: 860px;
  	margin: 0 auto;
  	padding: 30px;
    padding-bottom: 100px;
}

@media only screen and (min-width: 1400px) {
	#write {
		max-width: 1024px;
	}
}

@media only screen and (min-width: 1800px) {
	#write {
		max-width: 1200px;
	}
}

#write > ul:first-child,
#write > ol:first-child{
    margin-top: 30px;
}

a {
    color: #4183C4;
}
h1,
h2,
h3,
h4,
h5,
h6 {
    position: relative;
    margin-top: 1rem;
    margin-bottom: 1rem;
    font-weight: bold;
    line-height: 1.4;
    cursor: text;
}
h1:hover a.anchor,
h2:hover a.anchor,
h3:hover a.anchor,
h4:hover a.anchor,
h5:hover a.anchor,
h6:hover a.anchor {
    text-decoration: none;
}
h1 tt,
h1 code {
    font-size: inherit;
}
h2 tt,
h2 code {
    font-size: inherit;
}
h3 tt,
h3 code {
    font-size: inherit;
}
h4 tt,
h4 code {
    font-size: inherit;
}
h5 tt,
h5 code {
    font-size: inherit;
}
h6 tt,
h6 code {
    font-size: inherit;
}
h1 {
    font-size: 2.25em;
    line-height: 1.2;
    border-bottom: 1px solid #eee;
}
h2 {
    font-size: 1.75em;
    line-height: 1.225;
    border-bottom: 1px solid #eee;
}

/*@media print {
    .typora-export h1,
    .typora-export h2 {
        border-bottom: none;
        padding-bottom: initial;
    }

    .typora-export h1::after,
    .typora-export h2::after {
        content: "";
        display: block;
        height: 100px;
        margin-top: -96px;
        border-top: 1px solid #eee;
    }
}*/

h3 {
    font-size: 1.5em;
    line-height: 1.43;
}
h4 {
    font-size: 1.25em;
}
h5 {
    font-size: 1em;
}
h6 {
   font-size: 1em;
    color: #777;
}
p,
blockquote,
ul,
ol,
dl,
table{
    margin: 0.8em 0;
}
li>ol,
li>ul {
    margin: 0 0;
}
hr {
    height: 2px;
    padding: 0;
    margin: 16px 0;
    background-color: #e7e7e7;
    border: 0 none;
    overflow: hidden;
    box-sizing: content-box;
}

li p.first {
    display: inline-block;
}
ul,
ol {
    padding-left: 30px;
}
ul:first-child,
ol:first-child {
    margin-top: 0;
}
ul:last-child,
ol:last-child {
    margin-bottom: 0;
}
blockquote {
    border-left: 4px solid #dfe2e5;
    padding: 0 15px;
    color: #777777;
}
blockquote blockquote {
    padding-right: 0;
}
table {
    padding: 0;
    word-break: initial;
}
table tr {
    border: 1px solid #dfe2e5;
    margin: 0;
    padding: 0;
}
table tr:nth-child(2n),
thead {
    background-color: #f8f8f8;
}
table th {
    font-weight: bold;
    border: 1px solid #dfe2e5;
    border-bottom: 0;
    margin: 0;
    padding: 6px 13px;
}
table td {
    border: 1px solid #dfe2e5;
    margin: 0;
    padding: 6px 13px;
}
table th:first-child,
table td:first-child {
    margin-top: 0;
}
table th:last-child,
table td:last-child {
    margin-bottom: 0;
}

.CodeMirror-lines {
    padding-left: 4px;
}

.code-tooltip {
    box-shadow: 0 1px 1px 0 rgba(0,28,36,.3);
    border-top: 1px solid #eef2f2;
}

.md-fences,
code,
tt {
    border: 1px solid #e7eaed;
    background-color: #f8f8f8;
    border-radius: 3px;
    padding: 0;
    padding: 2px 4px 0px 4px;
    font-size: 0.9em;
}

code {
    background-color: #f3f4f4;
    padding: 0 2px 0 2px;
}

.md-fences {
    margin-bottom: 15px;
    margin-top: 15px;
    padding-top: 8px;
    padding-bottom: 6px;
}


.md-task-list-item > input {
  margin-left: -1.3em;
}

@media print {
    html {
        font-size: 13px;
    }
    pre {
        page-break-inside: avoid;
        word-wrap: break-word;
    }
}

.md-fences {
	background-color: #f8f8f8;
}
#write pre.md-meta-block {
	padding: 1rem;
    font-size: 85%;
    line-height: 1.45;
    background-color: #f7f7f7;
    border: 0;
    border-radius: 3px;
    color: #777777;
    margin-top: 0 !important;
}

.mathjax-block>.code-tooltip {
	bottom: .375rem;
}

.md-mathjax-midline {
    background: #fafafa;
}

#write>h3.md-focus:before{
	left: -1.5625rem;
	top: .375rem;
}
#write>h4.md-focus:before{
	left: -1.5625rem;
	top: .285714286rem;
}
#write>h5.md-focus:before{
	left: -1.5625rem;
	top: .285714286rem;
}
#write>h6.md-focus:before{
	left: -1.5625rem;
	top: .285714286rem;
}
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</style><title>Readme</title>
</head>
<body class='typora-export os-windows'><div class='typora-export-content'>
<div id='write'  class=''><p><img src="C:\Users\huangyijing\AppData\Roaming\Typora\typora-user-images\image-20241119210100900.png" referrerpolicy="no-referrer" alt="image-20241119210100900"></p><p><span>	</span><span>J积分的积分区域如上图所示。J积分计算公式：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n2" cid="n2" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-math-container"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="78.721ex" height="6.74ex" role="img" focusable="false" viewBox="0 -1733 34794.7 2978.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.819ex;"><defs><path id="MJX-629-TEX-I-1D43D" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 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xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>J</mi><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>M</mi></mrow></munderover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>w</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msup><mo>−</mo><msubsup><mi>σ</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msubsup><mfrac><mrow><mi>∂</mi><msubsup><mi>u</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msubsup></mrow><mrow><mi>∂</mi><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mrow></mfrac><mo>−</mo><msubsup><mi>σ</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msubsup><mfrac><mrow><mi>∂</mi><msubsup><mi>u</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msubsup></mrow><mrow><mi>∂</mi><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mrow></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mfrac><mrow><mi>d</mi><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msup></mrow><mrow><mi>d</mi><msub><mi>x</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mrow></mfrac><mo>−</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msubsup><mi>σ</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msubsup><mfrac><mrow><mi>∂</mi><msubsup><mi>u</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msubsup></mrow><mrow><mi>∂</mi><msub><mi>x</mi><mrow 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325T593 357T580 385Z"></path><path id="MJX-10-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D464" xlink:href="#MJX-10-TEX-I-1D464"></use></g><g data-mml-node="mi" transform="translate(749,363) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-10-TEX-I-1D45A"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>w</mi><mi>m</mi></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">w^m</script><span>表示第m个单元的应变能密度，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="11.879ex" height="1.414ex" role="img" focusable="false" viewBox="0 -431 5250.7 625" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-32-TEX-I-1D70E" d="M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z"></path><path id="MJX-32-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-32-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-32-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D70E" xlink:href="#MJX-32-TEX-I-1D70E"></use></g><g data-mml-node="TeXAtom" transform="translate(604,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-32-TEX-N-31"></use><use data-c="31" xlink:href="#MJX-32-TEX-N-31" transform="translate(500,0)"></use></g></g></g><g data-mml-node="mo" transform="translate(1361.1,0)"><use data-c="2C" xlink:href="#MJX-32-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1805.8,0)"><g data-mml-node="mi"><use data-c="1D70E" xlink:href="#MJX-32-TEX-I-1D70E"></use></g><g data-mml-node="TeXAtom" transform="translate(604,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-32-TEX-N-31"></use><use data-c="32" xlink:href="#MJX-32-TEX-N-32" transform="translate(500,0)"></use></g></g></g><g data-mml-node="mo" transform="translate(3166.9,0)"><use data-c="2C" xlink:href="#MJX-32-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(3611.5,0)"><g data-mml-node="mi"><use data-c="1D70E" xlink:href="#MJX-32-TEX-I-1D70E"></use></g><g data-mml-node="TeXAtom" transform="translate(604,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-32-TEX-N-32"></use><use data-c="32" xlink:href="#MJX-32-TEX-N-32" transform="translate(500,0)"></use></g></g></g><g data-mml-node="mo" transform="translate(4972.7,0)"><use data-c="2C" xlink:href="#MJX-32-TEX-N-2C"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub><mo>,</mo><msub><mi>σ</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub><mo>,</mo><msub><mi>σ</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub><mo>,</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">\sigma_{11},\sigma_{12},\sigma_{22},</script><span>分别表示第m个单元在方向1和方向2以及方向12上的单元应力, </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.57ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 2461.8 636" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-40-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-40-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-40-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-40-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-40-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-40-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1008.6,0)"><use data-c="2C" xlink:href="#MJX-40-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1453.2,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-40-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-40-TEX-N-32"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>x</mi><mn>2</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">x_1,x_2</script><span>表示单元在方向1和方向2上的直角坐标，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.57ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 2461.8 636" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-49-TEX-I-1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-49-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-49-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-49-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D462" xlink:href="#MJX-49-TEX-I-1D462"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-49-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1008.6,0)"><use data-c="2C" xlink:href="#MJX-49-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1453.2,0)"><g data-mml-node="mi"><use data-c="1D462" xlink:href="#MJX-49-TEX-I-1D462"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-49-TEX-N-32"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>,</mo><msub><mi>u</mi><mn>2</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">u_1,u_2</script><span>则表示单元在方向1和方向2上的位移。</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.466ex" height="1.977ex" role="img" focusable="false" viewBox="0 -716 1973.8 873.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-86-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-86-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-86-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-86-TEX-I-1D451"></use></g><g data-mml-node="msub" transform="translate(520,0)"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-86-TEX-I-1D434"></use></g><g data-mml-node="mi" transform="translate(783,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-86-TEX-I-1D45A"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msub><mi>A</mi><mi>m</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">dA_m</script><span>为第m个单元的面积, g为一个辅助函数，其具体的值与点的极坐标r有关,  计算公式为:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n14" cid="n14" mdtype="math_block" data-math-tag-before="0" data-math-labels="[]" data-math-tag-after="0"><div class="md-math-container"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="14.654ex" height="4.775ex" role="img" focusable="false" viewBox="0 -1259 6477.1 2110.6" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -1.927ex;"><defs><path id="MJX-630-TEX-I-1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path id="MJX-630-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-630-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-630-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-630-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-630-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-630-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-630-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D454" xlink:href="#MJX-630-TEX-I-1D454"></use></g><g data-mml-node="mo" transform="translate(477,0)"><use data-c="28" xlink:href="#MJX-630-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(866,0)"><use data-c="1D45F" xlink:href="#MJX-630-TEX-I-1D45F"></use></g><g data-mml-node="mo" transform="translate(1317,0)"><use data-c="29" xlink:href="#MJX-630-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(1983.8,0)"><use data-c="3D" xlink:href="#MJX-630-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(3039.6,0)"><g data-mml-node="mrow" transform="translate(438.3,676)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-630-TEX-I-1D45F"></use></g><g data-mml-node="mo" transform="translate(673.2,0)"><use data-c="2212" xlink:href="#MJX-630-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(1673.4,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-630-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-630-TEX-N-30"></use></g></g></g><g data-mml-node="mrow" transform="translate(220,-686)"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-630-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-630-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1109.8,0)"><use data-c="2212" xlink:href="#MJX-630-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(2110,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-630-TEX-I-1D45F"></use></g><g data-mml-node="mn" transform="translate(484,-150) scale(0.707)"><use data-c="30" 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287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-86-TEX-I-1D451"></use></g><g data-mml-node="msub" transform="translate(520,0)"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-86-TEX-I-1D434"></use></g><g data-mml-node="mi" transform="translate(783,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-86-TEX-I-1D45A"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msub><mi>A</mi><mi>m</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">dA_m</script><span>的计算方法:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n4" cid="n4" mdtype="math_block" data-math-tag-before="0" data-math-labels="[]" data-math-tag-after="0"><div 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322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-631-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-631-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-631-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-631-TEX-SO-5B" d="M202 -349V850H394V810H242V-309H394V-349H202Z"></path><path id="MJX-631-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-631-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 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data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="5B" xlink:href="#MJX-631-TEX-SO-5B"></use></g><g data-mml-node="mo" transform="translate(417,0)"><use data-c="28" xlink:href="#MJX-631-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(806,0)"><use data-c="1D45F" xlink:href="#MJX-631-TEX-I-1D45F"></use></g><g data-mml-node="mo" transform="translate(1479.2,0)"><use data-c="2B" xlink:href="#MJX-631-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(2479.4,0)"><use data-c="1D451" xlink:href="#MJX-631-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(2999.4,0)"><use data-c="1D45F" xlink:href="#MJX-631-TEX-I-1D45F"></use></g><g data-mml-node="msup" transform="translate(3450.4,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJX-631-TEX-N-29"></use></g><g data-mml-node="TeXAtom" transform="translate(422,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-631-TEX-N-32"></use></g></g></g><g data-mml-node="mo" transform="translate(4498.2,0)"><use data-c="2212" xlink:href="#MJX-631-TEX-N-2212"></use></g><g data-mml-node="msup" transform="translate(5498.4,0)"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-631-TEX-I-1D45F"></use></g><g data-mml-node="TeXAtom" transform="translate(484,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-631-TEX-N-32"></use></g></g></g><g data-mml-node="mo" transform="translate(6386,0) translate(0 -0.5)"><use data-c="5D" xlink:href="#MJX-631-TEX-SO-5D"></use></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(11086.4,0)"><g data-mml-node="mi"><use data-c="64" xlink:href="#MJX-631-TEX-N-64"></use></g></g><g data-mml-node="mi" transform="translate(11642.4,0)"><use data-c="1D703" xlink:href="#MJX-631-TEX-I-1D703"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><msup><mi>A</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mo stretchy="false">(</mo><mi>r</mi><mo>+</mo><mi>d</mi><mi>r</mi><msup><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>−</mo><msup><mi>r</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">]</mo></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>θ</mi></math></mjx-assistive-mml></mjx-container></div></div><p><img src="C:\Users\huangyijing\AppData\Roaming\Typora\typora-user-images\image-20241119211025817.png" referrerpolicy="no-referrer" alt="image-20241119211025817"></p><p><span>	</span><span>三个参数</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.467ex" height="2.034ex" role="img" focusable="false" viewBox="0 -705 3300.3 899" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-103-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-103-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path 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132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45F" xlink:href="#MJX-103-TEX-I-1D45F"></use></g><g data-mml-node="mo" transform="translate(451,0)"><use data-c="2C" xlink:href="#MJX-103-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(895.7,0)"><use data-c="1D451" xlink:href="#MJX-103-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(1415.7,0)"><use data-c="1D45F" xlink:href="#MJX-103-TEX-I-1D45F"></use></g><g data-mml-node="mo" transform="translate(1866.7,0)"><use data-c="2C" xlink:href="#MJX-103-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2311.3,0)"><use data-c="1D451" xlink:href="#MJX-103-TEX-I-1D451"></use></g><g data-mml-node="mi" transform="translate(2831.3,0)"><use data-c="1D703" xlink:href="#MJX-103-TEX-I-1D703"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>,</mo><mi>d</mi><mi>r</mi><mo>,</mo><mi>d</mi><mi>θ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">r, dr, d\theta</script><span>如上图所示.</span></p><p><span>	</span><span>接下来是应变能密度:</span></p><p><img src="C:\Users\huangyijing\AppData\Roaming\Typora\typora-user-images\image-20241119211126377.png" referrerpolicy="no-referrer" alt="image-20241119211126377"></p><p><span>	</span><span>上市中, </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.282ex" height="2.318ex" role="img" focusable="false" viewBox="0 -830.4 1450.8 1024.4" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-113-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-113-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-113-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-113-TEX-I-1D45D"></use></g><g data-mml-node="TeXAtom" transform="translate(536,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-113-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(345,0)"><use data-c="1D45A" xlink:href="#MJX-113-TEX-I-1D45A"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>m</mi></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">p^{im}</script><span>表示第m个单元中的第i个原子在变形之后的势能，相对应的，</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.282ex" height="2.583ex" role="img" focusable="false" viewBox="0 -830.4 1450.8 1141.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.704ex;"><defs><path id="MJX-117-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-117-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-117-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-117-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msubsup"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-117-TEX-I-1D45D"></use></g><g data-mml-node="TeXAtom" transform="translate(536,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-117-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(345,0)"><use data-c="1D45A" xlink:href="#MJX-117-TEX-I-1D45A"></use></g></g><g data-mml-node="mn" transform="translate(536,-295.7) scale(0.707)"><use data-c="30" xlink:href="#MJX-117-TEX-N-30"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>p</mi><mn>0</mn><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>m</mi></mrow></msubsup></math></mjx-assistive-mml></mjx-container><script type="math/tex">p^{im}_0</script><span>表示初始状态下的原子势能。</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.911ex" height="1.902ex" role="img" focusable="false" viewBox="0 -683 1286.8 840.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-121-TEX-I-1D449" d="M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z"></path><path id="MJX-121-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D449" xlink:href="#MJX-121-TEX-I-1D449"></use></g><g data-mml-node="mi" transform="translate(616,-150) scale(0.707)"><use data-c="1D45A" xlink:href="#MJX-121-TEX-I-1D45A"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>m</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">V_m</script><span>为第m个单元的体积，可以用上面计算出来的</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.466ex" height="1.977ex" role="img" focusable="false" viewBox="0 -716 1973.8 873.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-131-TEX-I-1D451" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path id="MJX-131-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path id="MJX-131-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D451" xlink:href="#MJX-131-TEX-I-1D451"></use></g><g data-mml-node="msub" transform="translate(520,0)"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-131-TEX-I-1D434"></use></g><g data-mml-node="TeXAtom" transform="translate(783,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-131-TEX-I-1D45A"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">dA_{m}</script><span>乘以样品的厚度计算获得。</span></p><p><span>	</span><span>接下来是计算单元应力：</span></p><p><img src="C:\Users\huangyijing\AppData\Roaming\Typora\typora-user-images\image-20241119211357937.png" referrerpolicy="no-referrer" alt="image-20241119211357937"></p><p><span>	</span><span>看公式很简单，就是把在单元内部的所有原子的应力分量加起来除以单元体积就可以获得。但是这里存在一个疑问，分子动力学获得的原子应力单位不是应力单位，而是应力乘以体积单位，所以这里的应力究竟指的是分子动力学里面的原子应力，还是指实际应力单位的应力呢？通过量纲分析，我倾向于认为这里的应力指的是原子应力，不过我换了一种平均化的方法，因为我提前在lammps的in文件中将原子应力除以了体积，所以获得的应力单位是GPa，就是真正的应力单位，所以我写程序的时候是直接把第m个单元里面的所有原子的应力分量加起来，再除以第m个单元内部的原子数量N，最终结果就代表了第m个单元内部的应力分量数值。</span></p><p><span>	</span><span>接下来我们需要计算每个单元四个角点的在水平和竖直方向上的位移分量:</span></p><p><img src="C:\Users\huangyijing\AppData\Roaming\Typora\typora-user-images\image-20241119211900028.png" referrerpolicy="no-referrer" alt="image-20241119211900028"></p><p><span>	</span><span>这里作者采用了空间平均的方法，找到距离每个单元的4个节点3埃米以内的所有原子，然后将每个坐落在对应角点距离内的原子质量</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.198ex" height="1.553ex" role="img" focusable="false" viewBox="0 -675.5 1413.5 686.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-141-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-141-TEX-I-1D6FC" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-141-TEX-I-1D45A"></use></g><g data-mml-node="TeXAtom" transform="translate(911,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D6FC" xlink:href="#MJX-141-TEX-I-1D6FC"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>m</mi><mrow data-mjx-texclass="ORD"><mi>α</mi></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">m^{\alpha}</script><span>乘以原子位移</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.506ex" height="1.553ex" role="img" focusable="false" viewBox="0 -675.5 1107.5 686.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-150-TEX-I-1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-150-TEX-I-1D6FC" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D462" xlink:href="#MJX-150-TEX-I-1D462"></use></g><g data-mml-node="TeXAtom" transform="translate(605,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D6FC" xlink:href="#MJX-150-TEX-I-1D6FC"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>u</mi><mrow data-mjx-texclass="ORD"><mi>α</mi></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">u^{\alpha}</script><span>, 将它们加起来，再除以这些原子的总质量，就得到了每个单位的4个节点的水平和竖直方向的位移。</span></p><p><span>	</span><span>接下来就是有限元分析中的的形函数:</span></p><p><img src="C:\Users\huangyijing\AppData\Roaming\Typora\typora-user-images\image-20241119212417733.png" referrerpolicy="no-referrer" alt="image-20241119212417733"></p><p><span>	</span><span>在二维有限元分析中，我们常常需要计算大量的二重积分，为了便于计算二次积分，我们需要将一个任意形状的四面体单元映射到一个标准的2x2正方形区域中，如下图所示:</span></p><p><img src="C:\Users\huangyijing\AppData\Roaming\Typora\typora-user-images\image-20241119213548047.png" referrerpolicy="no-referrer" alt="image-20241119213548047"></p><p><span>	</span><span>在上图中，作图中的坐标称为全局坐标或者直角坐标，用x和y表示，右图中的坐标系称为自然坐标系，用</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.991ex" height="2.057ex" role="img" focusable="false" viewBox="0 -704 438 909" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-153-TEX-I-1D709" d="M268 632Q268 704 296 704Q314 704 314 687Q314 682 311 664T308 635T309 620V616H315Q342 619 360 619Q443 619 443 586Q439 548 358 546H344Q326 546 317 549T290 566Q257 550 226 505T195 405Q195 381 201 364T211 342T218 337Q266 347 298 347Q375 347 375 314Q374 297 359 288T327 277T280 275Q234 275 208 283L195 286Q149 260 119 214T88 130Q88 116 90 108Q101 79 129 63T229 20Q238 17 243 15Q337 -21 354 -33Q383 -53 383 -94Q383 -137 351 -171T273 -205Q240 -205 202 -190T158 -167Q156 -163 156 -159Q156 -151 161 -146T176 -140Q182 -140 189 -143Q232 -168 274 -168Q286 -168 292 -165Q313 -151 313 -129Q313 -112 301 -104T232 -75Q214 -68 204 -64Q198 -62 171 -52T136 -38T107 -24T78 -8T56 12T36 37T26 66T21 103Q21 149 55 206T145 301L154 307L148 313Q141 319 136 323T124 338T111 358T103 382T99 413Q99 471 143 524T259 602L271 607Q268 618 268 632Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D709" xlink:href="#MJX-153-TEX-I-1D709"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ξ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\xi</script><span>和</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.124ex" height="1.489ex" role="img" focusable="false" viewBox="0 -442 497 658" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.489ex;"><defs><path id="MJX-405-TEX-I-1D702" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q156 442 175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336V326Q503 302 439 53Q381 -182 377 -189Q364 -216 332 -216Q319 -216 310 -208T299 -186Q299 -177 358 57L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D702" xlink:href="#MJX-405-TEX-I-1D702"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>η</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\eta</script><span>表示。它们之间存在着如下的映射函数关系:</span></p><div 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transform="translate(836,-150) scale(0.707)"><use data-c="34" xlink:href="#MJX-233-TEX-N-34"></use></g></g><g data-mml-node="mo" transform="translate(1239.6,0)"><use data-c="28" xlink:href="#MJX-233-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1628.6,0)"><use data-c="1D709" xlink:href="#MJX-233-TEX-I-1D709"></use></g><g data-mml-node="mo" transform="translate(2066.6,0)"><use data-c="2C" xlink:href="#MJX-233-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2511.2,0)"><use data-c="1D702" xlink:href="#MJX-233-TEX-I-1D702"></use></g><g data-mml-node="mo" transform="translate(3008.2,0)"><use data-c="29" xlink:href="#MJX-233-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mn>4</mn></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">N_4(\xi, 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transform="translate(8609.2,0)"><use data-c="28" xlink:href="#MJX-633-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(8998.2,0)"><use data-c="31" xlink:href="#MJX-633-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(9720.4,0)"><use data-c="2B" xlink:href="#MJX-633-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(10720.7,0)"><use data-c="1D702" xlink:href="#MJX-633-TEX-I-1D702"></use></g><g data-mml-node="mo" transform="translate(11217.7,0)"><use data-c="29" xlink:href="#MJX-633-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(11606.7,0)"><use data-c="2C" xlink:href="#MJX-633-TEX-N-2C"></use></g><g data-mml-node="mfrac" transform="translate(12051.3,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mi"><use data-c="1D715" xlink:href="#MJX-633-TEX-I-1D715"></use></g><g data-mml-node="mi" transform="translate(566,0)"><use data-c="1D441" xlink:href="#MJX-633-TEX-I-1D441"></use></g><g data-mml-node="mn" 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transform="translate(1454,0)"><use data-c="31" xlink:href="#MJX-633-TEX-N-31"></use></g></g><g data-mml-node="mrow" transform="translate(665.5,-686)"><g data-mml-node="mi"><use data-c="1D715" xlink:href="#MJX-633-TEX-I-1D715"></use></g><g data-mml-node="mi" transform="translate(566,0)"><use data-c="1D702" xlink:href="#MJX-633-TEX-I-1D702"></use></g></g><rect width="2154" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(23610.8,0)"><use data-c="3D" xlink:href="#MJX-633-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(24666.6,0)"><g data-mml-node="mn" transform="translate(220,676)"><use data-c="31" xlink:href="#MJX-633-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-686)"><use data-c="34" xlink:href="#MJX-633-TEX-N-34"></use></g><rect width="700" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(25606.6,0)"><use data-c="28" xlink:href="#MJX-633-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(25995.6,0)"><use data-c="31" xlink:href="#MJX-633-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(26717.8,0)"><use data-c="2212" xlink:href="#MJX-633-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(27718,0)"><use data-c="1D709" xlink:href="#MJX-633-TEX-I-1D709"></use></g><g data-mml-node="mo" transform="translate(28156,0)"><use data-c="29" xlink:href="#MJX-633-TEX-N-29"></use></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing=".5em" columnspacing="1em" displaystyle="true"><mtr><mtd><msub><mi>N</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>η</mi><mo stretchy="false">)</mo><mo>,</mo><mfrac><mrow><mi>∂</mi><mi>N</mi><mn>1</mn></mrow><mrow><mi>∂</mi><mi>ξ</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>η</mi><mo stretchy="false">)</mo><mo>,</mo><mfrac><mrow><mi>∂</mi><mi>N</mi><mn>1</mn></mrow><mrow><mi>∂</mi><mi>η</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>ξ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mi>N</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>η</mi><mo stretchy="false">)</mo><mo>,</mo><mfrac><mrow><mi>∂</mi><mi>N</mi><mn>1</mn></mrow><mrow><mi>∂</mi><mi>ξ</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>η</mi><mo stretchy="false">)</mo><mo>,</mo><mfrac><mrow><mi>∂</mi><mi>N</mi><mn>1</mn></mrow><mrow><mi>∂</mi><mi>η</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>ξ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mi>N</mi><mn>3</mn></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>η</mi><mo stretchy="false">)</mo><mo>,</mo><mfrac><mrow><mi>∂</mi><mi>N</mi><mn>1</mn></mrow><mrow><mi>∂</mi><mi>ξ</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>η</mi><mo stretchy="false">)</mo><mo>,</mo><mfrac><mrow><mi>∂</mi><mi>N</mi><mn>1</mn></mrow><mrow><mi>∂</mi><mi>η</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>ξ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mi>N</mi><mn>4</mn></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>η</mi><mo stretchy="false">)</mo><mo>,</mo><mfrac><mrow><mi>∂</mi><mi>N</mi><mn>1</mn></mrow><mrow><mi>∂</mi><mi>ξ</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>η</mi><mo stretchy="false">)</mo><mo>,</mo><mfrac><mrow><mi>∂</mi><mi>N</mi><mn>1</mn></mrow><mrow><mi>∂</mi><mi>η</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>ξ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>	</span><span>可以看到这四个形函数和文献中描述的公式（8）形式一致。接下来我们需要计算原子的应变和辅助函数g的值：</span></p><p><img src="C:\Users\huangyijing\AppData\Roaming\Typora\typora-user-images\image-20241119213118553.png" referrerpolicy="no-referrer" alt="image-20241119213118553"></p><p><span>	</span><span>首先从上图的第一行可以看到，原子在</span><strong><span>j</span></strong><span>方向上的位移</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.054ex" height="1.072ex" role="img" focusable="false" viewBox="0 -452 466 474" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-254-TEX-I-1D700" d="M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D700" xlink:href="#MJX-254-TEX-I-1D700"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ε</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\varepsilon</script><span>等于单元的4个节点在j方向上的位移</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.438ex" height="2.866ex" role="img" focusable="false" viewBox="0 -830.4 1519.8 1266.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.987ex;"><defs><path id="MJX-264-TEX-I-1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-264-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-264-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-264-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" 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xmlns="http://www.w3.org/2000/svg" width="2.556ex" height="1.902ex" role="img" focusable="false" viewBox="0 -683 1130 840.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-267-TEX-I-1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z"></path><path id="MJX-267-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D441" xlink:href="#MJX-267-TEX-I-1D441"></use></g><g data-mml-node="mi" transform="translate(836,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-267-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">N_i</script><span>对</span><strong><span>x1</span></strong><span>的偏导数之和。其中单元的4个节点在j方向上的位移</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.438ex" height="2.866ex" role="img" focusable="false" viewBox="0 -830.4 1519.8 1266.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.987ex;"><defs><path id="MJX-264-TEX-I-1D462" d="M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-264-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-264-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-264-TEX-I-1D457" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msubsup"><g data-mml-node="mi"><use data-c="1D462" xlink:href="#MJX-264-TEX-I-1D462"></use></g><g data-mml-node="TeXAtom" transform="translate(605,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-264-TEX-I-1D456"></use></g><g data-mml-node="mi" transform="translate(345,0)"><use data-c="1D45A" xlink:href="#MJX-264-TEX-I-1D45A"></use></g></g><g data-mml-node="mi" transform="translate(605,-292.2) scale(0.707)"><use data-c="1D457" xlink:href="#MJX-264-TEX-I-1D457"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>u</mi><mi>j</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>m</mi></mrow></msubsup></math></mjx-assistive-mml></mjx-container><script type="math/tex">u_j^{im}</script><span>在公式（7）中已经解决，辅助函数g只需要把直角坐标换成极坐标之后，也只是简单的四则运算而已，难点在于后续的形函数</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.891ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3045.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-563-TEX-I-1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z"></path><path id="MJX-563-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-563-TEX-I-1D709" d="M268 632Q268 704 296 704Q314 704 314 687Q314 682 311 664T308 635T309 620V616H315Q342 619 360 619Q443 619 443 586Q439 548 358 546H344Q326 546 317 549T290 566Q257 550 226 505T195 405Q195 381 201 364T211 342T218 337Q266 347 298 347Q375 347 375 314Q374 297 359 288T327 277T280 275Q234 275 208 283L195 286Q149 260 119 214T88 130Q88 116 90 108Q101 79 129 63T229 20Q238 17 243 15Q337 -21 354 -33Q383 -53 383 -94Q383 -137 351 -171T273 -205Q240 -205 202 -190T158 -167Q156 -163 156 -159Q156 -151 161 -146T176 -140Q182 -140 189 -143Q232 -168 274 -168Q286 -168 292 -165Q313 -151 313 -129Q313 -112 301 -104T232 -75Q214 -68 204 -64Q198 -62 171 -52T136 -38T107 -24T78 -8T56 12T36 37T26 66T21 103Q21 149 55 206T145 301L154 307L148 313Q141 319 136 323T124 338T111 358T103 382T99 413Q99 471 143 524T259 602L271 607Q268 618 268 632Z"></path><path id="MJX-563-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-563-TEX-I-1D702" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q156 442 175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336V326Q503 302 439 53Q381 -182 377 -189Q364 -216 332 -216Q319 -216 310 -208T299 -186Q299 -177 358 57L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-563-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D441" xlink:href="#MJX-563-TEX-I-1D441"></use></g><g data-mml-node="mo" transform="translate(888,0)"><use data-c="28" xlink:href="#MJX-563-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1277,0)"><use data-c="1D709" xlink:href="#MJX-563-TEX-I-1D709"></use></g><g data-mml-node="mo" transform="translate(1715,0)"><use data-c="2C" xlink:href="#MJX-563-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2159.7,0)"><use data-c="1D702" xlink:href="#MJX-563-TEX-I-1D702"></use></g><g data-mml-node="mo" transform="translate(2656.7,0)"><use data-c="29" xlink:href="#MJX-563-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">N(\xi,\eta)</script><span>偏导。首先我们需要明确，形函数N是自然坐标</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.991ex" height="2.057ex" role="img" focusable="false" viewBox="0 -704 438 909" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-153-TEX-I-1D709" d="M268 632Q268 704 296 704Q314 704 314 687Q314 682 311 664T308 635T309 620V616H315Q342 619 360 619Q443 619 443 586Q439 548 358 546H344Q326 546 317 549T290 566Q257 550 226 505T195 405Q195 381 201 364T211 342T218 337Q266 347 298 347Q375 347 375 314Q374 297 359 288T327 277T280 275Q234 275 208 283L195 286Q149 260 119 214T88 130Q88 116 90 108Q101 79 129 63T229 20Q238 17 243 15Q337 -21 354 -33Q383 -53 383 -94Q383 -137 351 -171T273 -205Q240 -205 202 -190T158 -167Q156 -163 156 -159Q156 -151 161 -146T176 -140Q182 -140 189 -143Q232 -168 274 -168Q286 -168 292 -165Q313 -151 313 -129Q313 -112 301 -104T232 -75Q214 -68 204 -64Q198 -62 171 -52T136 -38T107 -24T78 -8T56 12T36 37T26 66T21 103Q21 149 55 206T145 301L154 307L148 313Q141 319 136 323T124 338T111 358T103 382T99 413Q99 471 143 524T259 602L271 607Q268 618 268 632Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D709" xlink:href="#MJX-153-TEX-I-1D709"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ξ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\xi</script><span>和</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.124ex" height="1.489ex" role="img" focusable="false" viewBox="0 -442 497 658" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.489ex;"><defs><path id="MJX-405-TEX-I-1D702" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q156 442 175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336V326Q503 302 439 53Q381 -182 377 -189Q364 -216 332 -216Q319 -216 310 -208T299 -186Q299 -177 358 57L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D702" xlink:href="#MJX-405-TEX-I-1D702"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>η</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\eta</script><span>的函数，但是自然坐标又是全局坐标</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-279-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-279-TEX-I-1D465"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">x</script><span>和</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-280-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-280-TEX-I-1D466"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">y</script><span>的函数, 即:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n62" cid="n62" mdtype="math_block" data-math-tag-before="0" data-math-labels="[]" data-math-tag-after="0"><div class="md-math-container"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="10.435ex" height="5.656ex" role="img" focusable="false" viewBox="0 -1500 4612.2 2500" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.262ex;"><defs><path id="MJX-634-TEX-I-1D709" d="M268 632Q268 704 296 704Q314 704 314 687Q314 682 311 664T308 635T309 620V616H315Q342 619 360 619Q443 619 443 586Q439 548 358 546H344Q326 546 317 549T290 566Q257 550 226 505T195 405Q195 381 201 364T211 342T218 337Q266 347 298 347Q375 347 375 314Q374 297 359 288T327 277T280 275Q234 275 208 283L195 286Q149 260 119 214T88 130Q88 116 90 108Q101 79 129 63T229 20Q238 17 243 15Q337 -21 354 -33Q383 -53 383 -94Q383 -137 351 -171T273 -205Q240 -205 202 -190T158 -167Q156 -163 156 -159Q156 -151 161 -146T176 -140Q182 -140 189 -143Q232 -168 274 -168Q286 -168 292 -165Q313 -151 313 -129Q313 -112 301 -104T232 -75Q214 -68 204 -64Q198 -62 171 -52T136 -38T107 -24T78 -8T56 12T36 37T26 66T21 103Q21 149 55 206T145 301L154 307L148 313Q141 319 136 323T124 338T111 358T103 382T99 413Q99 471 143 524T259 602L271 607Q268 618 268 632Z"></path><path id="MJX-634-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 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xlink:href="#MJX-634-TEX-N-29"></use></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing=".5em" columnspacing="1em" displaystyle="true"><mtr><mtd><mi>ξ</mi><mo>=</mo><mi>ξ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>η</mi><mo>=</mo><mi>η</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>	</span><span>所以形函数也可以是全局坐标x和y的函数， 即</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="17.086ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 7551.9 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-326-TEX-I-1D441" d="M234 637Q231 637 226 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412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D441" xlink:href="#MJX-326-TEX-I-1D441"></use></g><g data-mml-node="mo" transform="translate(888,0)"><use data-c="28" xlink:href="#MJX-326-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1277,0)"><use data-c="1D709" xlink:href="#MJX-326-TEX-I-1D709"></use></g><g data-mml-node="mo" transform="translate(1715,0)"><use 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xlink:href="#MJX-636-TEX-S4-5B"></use></g><g data-mml-node="mtable" transform="translate(583,0)"><g data-mml-node="mtr" transform="translate(0,875.8)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(220,394) scale(0.707)"><g data-mml-node="mi"><use data-c="1D715" xlink:href="#MJX-636-TEX-I-1D715"></use></g><g data-mml-node="mi" transform="translate(566,0)"><use data-c="1D441" xlink:href="#MJX-636-TEX-I-1D441"></use></g></g><g data-mml-node="mrow" transform="translate(331.7,-345.6) scale(0.707)"><g data-mml-node="mi"><use data-c="1D715" xlink:href="#MJX-636-TEX-I-1D715"></use></g><g data-mml-node="mi" transform="translate(566,0)"><use data-c="1D465" xlink:href="#MJX-636-TEX-I-1D465"></use></g></g><rect width="1228.1" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0,-784.9)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(220,394) scale(0.707)"><g 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rowspacing="4pt"><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>N</mi></mrow><mrow><mi>∂</mi><mi>ξ</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>N</mi></mrow><mrow><mi>∂</mi><mi>η</mi></mrow></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>y</mi></mrow><mrow><mi>∂</mi><mi>ξ</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mi>∂</mi><mi>x</mi></mrow><mrow><mi>∂</mi><mi>ξ</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>x</mi></mrow><mrow><mi>∂</mi><mi>η</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mi>∂</mi><mi>y</mi></mrow><mrow><mi>∂</mi><mi>η</mi></mrow></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>N</mi></mrow><mrow><mi>∂</mi><mi>x</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>N</mi></mrow><mrow><mi>∂</mi><mi>y</mi></mrow></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container></div></div><p><span>	</span><span>其中矩阵:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n79" cid="n79" mdtype="math_block" data-math-tag-before="0" data-math-labels="[]" data-math-tag-after="0"><div class="md-math-container"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="10.793ex" height="7.624ex" role="img" focusable="false" viewBox="0 -1935 4770.3 3369.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -3.247ex;"><defs><path id="MJX-637-TEX-S4-23A1" d="M319 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transform="translate(4103.3,0)"><use data-c="23A4" xlink:href="#MJX-637-TEX-S4-23A4" transform="translate(0,781)"></use><use data-c="23A6" xlink:href="#MJX-637-TEX-S4-23A6" transform="translate(0,-791)"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>x</mi></mrow><mrow><mi>∂</mi><mi>ξ</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mi>∂</mi><mi>y</mi></mrow><mrow><mi>∂</mi><mi>ξ</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>x</mi></mrow><mrow><mi>∂</mi><mi>η</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mi>∂</mi><mi>y</mi></mrow><mrow><mi>∂</mi><mi>η</mi></mrow></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container></div></div><p><span>	</span><span>被称为雅可比矩阵</span><strong><span>J</span></strong><span>，这个矩阵与文献中的公式(10)完全一致：</span></p><p><img src="C:\Users\huangyijing\AppData\Roaming\Typora\typora-user-images\image-20241119220206737.png" referrerpolicy="no-referrer" alt="image-20241119220206737"><span>而根据上面的公式，我们发现只要知道了自然坐标</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.991ex" height="2.057ex" role="img" focusable="false" viewBox="0 -704 438 909" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-153-TEX-I-1D709" d="M268 632Q268 704 296 704Q314 704 314 687Q314 682 311 664T308 635T309 620V616H315Q342 619 360 619Q443 619 443 586Q439 548 358 546H344Q326 546 317 549T290 566Q257 550 226 505T195 405Q195 381 201 364T211 342T218 337Q266 347 298 347Q375 347 375 314Q374 297 359 288T327 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transform="translate(605,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-638-TEX-N-32"></use></g></g><g data-mml-node="mn" transform="translate(474.3,-686)"><use data-c="34" xlink:href="#MJX-638-TEX-N-34"></use></g><rect width="1208.6" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(8950.8,0)"><use data-c="28" xlink:href="#MJX-638-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(9339.8,0)"><use data-c="31" xlink:href="#MJX-638-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(10062,0)"><use data-c="2212" xlink:href="#MJX-638-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(11062.2,0)"><use data-c="1D702" xlink:href="#MJX-638-TEX-I-1D702"></use></g><g data-mml-node="mo" transform="translate(11559.2,0)"><use data-c="29" xlink:href="#MJX-638-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(12170.4,0)"><use data-c="2B" xlink:href="#MJX-638-TEX-N-2B"></use></g><g data-mml-node="mfrac" transform="translate(13170.7,0)"><g data-mml-node="msub" transform="translate(220,676)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-638-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="33" xlink:href="#MJX-638-TEX-N-33"></use></g></g><g data-mml-node="mn" transform="translate(474.3,-686)"><use data-c="34" xlink:href="#MJX-638-TEX-N-34"></use></g><rect width="1208.6" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(14619.2,0)"><use data-c="28" xlink:href="#MJX-638-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(15008.2,0)"><use data-c="31" xlink:href="#MJX-638-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(15730.4,0)"><use data-c="2B" xlink:href="#MJX-638-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(16730.7,0)"><use data-c="1D702" xlink:href="#MJX-638-TEX-I-1D702"></use></g><g data-mml-node="mo" transform="translate(17227.7,0)"><use data-c="29" xlink:href="#MJX-638-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(17838.9,0)"><use data-c="2212" xlink:href="#MJX-638-TEX-N-2212"></use></g><g data-mml-node="mfrac" transform="translate(18839.1,0)"><g data-mml-node="msub" transform="translate(220,676)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-638-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="34" xlink:href="#MJX-638-TEX-N-34"></use></g></g><g data-mml-node="mn" transform="translate(474.3,-686)"><use data-c="34" xlink:href="#MJX-638-TEX-N-34"></use></g><rect width="1208.6" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(20287.7,0)"><use data-c="28" xlink:href="#MJX-638-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(20676.7,0)"><use data-c="31" xlink:href="#MJX-638-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(21398.9,0)"><use data-c="2B" xlink:href="#MJX-638-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(22399.1,0)"><use data-c="1D702" xlink:href="#MJX-638-TEX-I-1D702"></use></g><g data-mml-node="mo" transform="translate(22896.1,0)"><use data-c="29" xlink:href="#MJX-638-TEX-N-29"></use></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing=".5em" columnspacing="1em" displaystyle="true"><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>x</mi></mrow><mrow><mi>∂</mi><mi>ξ</mi></mrow></mfrac><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mfrac><mrow><mi>∂</mi><msub><mi>N</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><mrow><mi>∂</mi><mi>ξ</mi></mrow></mfrac><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mfrac><mrow><mi>∂</mi><msub><mi>N</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><mrow><mi>∂</mi><mi>ξ</mi></mrow></mfrac><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub><mfrac><mrow><mi>∂</mi><msub><mi>N</mi><mn>3</mn></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><mrow><mi>∂</mi><mi>ξ</mi></mrow></mfrac><mo>+</mo><msub><mi>x</mi><mn>4</mn></msub><mfrac><mrow><mi>∂</mi><msub><mi>N</mi><mn>4</mn></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><mrow><mi>∂</mi><mi>ξ</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mo>=</mo><mo>−</mo><mfrac><msub><mi>x</mi><mn>1</mn></msub><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>η</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><msub><mi>x</mi><mn>2</mn></msub><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>η</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><msub><mi>x</mi><mn>3</mn></msub><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>η</mi><mo stretchy="false">)</mo><mo>−</mo><mfrac><msub><mi>x</mi><mn>4</mn></msub><mn>4</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>η</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p>&nbsp;</p><p><span>	</span><span>同时我们发现，如果已知自然坐标</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.991ex" height="2.057ex" role="img" focusable="false" viewBox="0 -704 438 909" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-153-TEX-I-1D709" d="M268 632Q268 704 296 704Q314 704 314 687Q314 682 311 664T308 635T309 620V616H315Q342 619 360 619Q443 619 443 586Q439 548 358 546H344Q326 546 317 549T290 566Q257 550 226 505T195 405Q195 381 201 364T211 342T218 337Q266 347 298 347Q375 347 375 314Q374 297 359 288T327 277T280 275Q234 275 208 283L195 286Q149 260 119 214T88 130Q88 116 90 108Q101 79 129 63T229 20Q238 17 243 15Q337 -21 354 -33Q383 -53 383 -94Q383 -137 351 -171T273 -205Q240 -205 202 -190T158 -167Q156 -163 156 -159Q156 -151 161 -146T176 -140Q182 -140 189 -143Q232 -168 274 -168Q286 -168 292 -165Q313 -151 313 -129Q313 -112 301 -104T232 -75Q214 -68 204 -64Q198 -62 171 -52T136 -38T107 -24T78 -8T56 12T36 37T26 66T21 103Q21 149 55 206T145 301L154 307L148 313Q141 319 136 323T124 338T111 358T103 382T99 413Q99 471 143 524T259 602L271 607Q268 618 268 632Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D709" xlink:href="#MJX-153-TEX-I-1D709"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ξ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\xi</script><span>和</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.124ex" height="1.489ex" role="img" focusable="false" viewBox="0 -442 497 658" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.489ex;"><defs><path id="MJX-405-TEX-I-1D702" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q156 442 175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336V326Q503 302 439 53Q381 -182 377 -189Q364 -216 332 -216Q319 -216 310 -208T299 -186Q299 -177 358 57L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D702" xlink:href="#MJX-405-TEX-I-1D702"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>η</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\eta</script><span>，求出来雅可比矩阵的逆矩阵</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.786ex" height="1.937ex" role="img" focusable="false" viewBox="0 -833.9 1673.6 855.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-412-TEX-I-1D43D" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path id="MJX-412-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-412-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D43D" xlink:href="#MJX-412-TEX-I-1D43D"></use></g><g data-mml-node="TeXAtom" transform="translate(719.9,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-412-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJX-412-TEX-N-31"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>J</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">J^{-1}</script><span>之后, 可以得到如下等式:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n83" cid="n83" mdtype="math_block" data-math-tag-before="0" data-math-labels="[]" data-math-tag-after="0"><div class="md-math-container"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="19.1ex" height="7.213ex" role="img" focusable="false" viewBox="0 -1844 8442.1 3188" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -3.041ex;"><defs><path id="MJX-639-TEX-S4-5B" d="M269 -1249V1750H577V1677H342V-1176H577V-1249H269Z"></path><path id="MJX-639-TEX-I-1D715" d="M202 508Q179 508 169 520T158 547Q158 557 164 577T185 624T230 675T301 710L333 715H345Q378 715 384 714Q447 703 489 661T549 568T566 457Q566 362 519 240T402 53Q321 -22 223 -22Q123 -22 73 56Q42 102 42 148V159Q42 276 129 370T322 465Q383 465 414 434T455 367L458 378Q478 461 478 515Q478 603 437 639T344 676Q266 676 223 612Q264 606 264 572Q264 547 246 528T202 508ZM430 306Q430 372 401 400T333 428Q270 428 222 382Q197 354 183 323T150 221Q132 149 132 116Q132 21 232 21Q244 21 250 22Q327 35 374 112Q389 137 409 196T430 306Z"></path><path id="MJX-639-TEX-I-1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 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173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-639-TEX-S4-5D" d="M5 1677V1750H313V-1249H5V-1176H240V1677H5Z"></path><path id="MJX-639-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-639-TEX-I-1D43D" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path id="MJX-639-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-639-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-639-TEX-I-1D709" d="M268 632Q268 704 296 704Q314 704 314 687Q314 682 311 664T308 635T309 620V616H315Q342 619 360 619Q443 619 443 586Q439 548 358 546H344Q326 546 317 549T290 566Q257 550 226 505T195 405Q195 381 201 364T211 342T218 337Q266 347 298 347Q375 347 375 314Q374 297 359 288T327 277T280 275Q234 275 208 283L195 286Q149 260 119 214T88 130Q88 116 90 108Q101 79 129 63T229 20Q238 17 243 15Q337 -21 354 -33Q383 -53 383 -94Q383 -137 351 -171T273 -205Q240 -205 202 -190T158 -167Q156 -163 156 -159Q156 -151 161 -146T176 -140Q182 -140 189 -143Q232 -168 274 -168Q286 -168 292 -165Q313 -151 313 -129Q313 -112 301 -104T232 -75Q214 -68 204 -64Q198 -62 171 -52T136 -38T107 -24T78 -8T56 12T36 37T26 66T21 103Q21 149 55 206T145 301L154 307L148 313Q141 319 136 323T124 338T111 358T103 382T99 413Q99 471 143 524T259 602L271 607Q268 618 268 632Z"></path><path id="MJX-639-TEX-I-1D702" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q156 442 175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336V326Q503 302 439 53Q381 -182 377 -189Q364 -216 332 -216Q319 -216 310 -208T299 -186Q299 -177 358 57L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 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xlink:href="#MJX-639-TEX-S4-5D"></use></g></g><g data-mml-node="mo" transform="translate(2911.9,0)"><use data-c="3D" xlink:href="#MJX-639-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(3967.7,0)"><g data-mml-node="mi"><use data-c="1D43D" xlink:href="#MJX-639-TEX-I-1D43D"></use></g><g data-mml-node="TeXAtom" transform="translate(719.9,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-639-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJX-639-TEX-N-31"></use></g></g></g><g data-mml-node="mrow" transform="translate(5807.9,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="5B" xlink:href="#MJX-639-TEX-S4-5B"></use></g><g data-mml-node="mtable" transform="translate(583,0)"><g data-mml-node="mtr" transform="translate(0,944.4)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(220,394) scale(0.707)"><g data-mml-node="mi"><use data-c="1D715" xlink:href="#MJX-639-TEX-I-1D715"></use></g><g data-mml-node="mi" transform="translate(566,0)"><use data-c="1D441" xlink:href="#MJX-639-TEX-I-1D441"></use></g></g><g data-mml-node="mrow" transform="translate(379.1,-345.6) scale(0.707)"><g data-mml-node="mi"><use data-c="1D715" xlink:href="#MJX-639-TEX-I-1D715"></use></g><g data-mml-node="mi" transform="translate(566,0)"><use data-c="1D709" xlink:href="#MJX-639-TEX-I-1D709"></use></g></g><rect width="1228.1" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mtr" transform="translate(0,-845.7)"><g data-mml-node="mtd"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(220,394) scale(0.707)"><g data-mml-node="mi"><use data-c="1D715" xlink:href="#MJX-639-TEX-I-1D715"></use></g><g data-mml-node="mi" transform="translate(566,0)"><use data-c="1D441" xlink:href="#MJX-639-TEX-I-1D441"></use></g></g><g data-mml-node="mrow" transform="translate(358.2,-345.6) scale(0.707)"><g data-mml-node="mi"><use data-c="1D715" xlink:href="#MJX-639-TEX-I-1D715"></use></g><g data-mml-node="mi" transform="translate(566,0)"><use data-c="1D702" xlink:href="#MJX-639-TEX-I-1D702"></use></g></g><rect width="1228.1" height="60" x="120" y="220"></rect></g></g></g></g><g data-mml-node="mo" transform="translate(2051.1,0) translate(0 -0.5)"><use data-c="5D" xlink:href="#MJX-639-TEX-S4-5D"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>N</mi></mrow><mrow><mi>∂</mi><mi>x</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>N</mi></mrow><mrow><mi>∂</mi><mi>y</mi></mrow></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><msup><mi>J</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>N</mi></mrow><mrow><mi>∂</mi><mi>ξ</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>N</mi></mrow><mrow><mi>∂</mi><mi>η</mi></mrow></mfrac></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container></div></div><p><span>	</span><span>而4个形函数</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.891ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3045.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-439-TEX-I-1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 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xlink:href="#MJX-439-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1277,0)"><use data-c="1D709" xlink:href="#MJX-439-TEX-I-1D709"></use></g><g data-mml-node="mo" transform="translate(1715,0)"><use data-c="2C" xlink:href="#MJX-439-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2159.7,0)"><use data-c="1D702" xlink:href="#MJX-439-TEX-I-1D702"></use></g><g data-mml-node="mo" transform="translate(2656.7,0)"><use data-c="29" xlink:href="#MJX-439-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">N(\xi, \eta)</script><span>关于自然坐标</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.991ex" height="2.057ex" role="img" focusable="false" viewBox="0 -704 438 909" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-153-TEX-I-1D709" d="M268 632Q268 704 296 704Q314 704 314 687Q314 682 311 664T308 635T309 620V616H315Q342 619 360 619Q443 619 443 586Q439 548 358 546H344Q326 546 317 549T290 566Q257 550 226 505T195 405Q195 381 201 364T211 342T218 337Q266 347 298 347Q375 347 375 314Q374 297 359 288T327 277T280 275Q234 275 208 283L195 286Q149 260 119 214T88 130Q88 116 90 108Q101 79 129 63T229 20Q238 17 243 15Q337 -21 354 -33Q383 -53 383 -94Q383 -137 351 -171T273 -205Q240 -205 202 -190T158 -167Q156 -163 156 -159Q156 -151 161 -146T176 -140Q182 -140 189 -143Q232 -168 274 -168Q286 -168 292 -165Q313 -151 313 -129Q313 -112 301 -104T232 -75Q214 -68 204 -64Q198 -62 171 -52T136 -38T107 -24T78 -8T56 12T36 37T26 66T21 103Q21 149 55 206T145 301L154 307L148 313Q141 319 136 323T124 338T111 358T103 382T99 413Q99 471 143 524T259 602L271 607Q268 618 268 632Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D709" xlink:href="#MJX-153-TEX-I-1D709"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ξ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\xi</script><span>和</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.124ex" height="1.489ex" role="img" focusable="false" viewBox="0 -442 497 658" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.489ex;"><defs><path id="MJX-405-TEX-I-1D702" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q156 442 175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336V326Q503 302 439 53Q381 -182 377 -189Q364 -216 332 -216Q319 -216 310 -208T299 -186Q299 -177 358 57L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D702" xlink:href="#MJX-405-TEX-I-1D702"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>η</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\eta</script><span>的表达式以及对应偏导我们都已经在上文中展示过了，所以利用上述等式我们可以轻松求出来4个形函数</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.891ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3045.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-439-TEX-I-1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z"></path><path id="MJX-439-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-439-TEX-I-1D709" d="M268 632Q268 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data-mml-node="math"><g data-mml-node="mi"><use data-c="1D441" xlink:href="#MJX-439-TEX-I-1D441"></use></g><g data-mml-node="mo" transform="translate(888,0)"><use data-c="28" xlink:href="#MJX-439-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1277,0)"><use data-c="1D709" xlink:href="#MJX-439-TEX-I-1D709"></use></g><g data-mml-node="mo" transform="translate(1715,0)"><use data-c="2C" xlink:href="#MJX-439-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2159.7,0)"><use data-c="1D702" xlink:href="#MJX-439-TEX-I-1D702"></use></g><g data-mml-node="mo" transform="translate(2656.7,0)"><use data-c="29" xlink:href="#MJX-439-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">N(\xi, \eta)</script><span>对全局坐标x和y的偏导。</span></p><p><span>	</span><span>那么现在让我们回顾一下单元应变和单元辅助函数g：</span></p><p><img src="C:\Users\huangyijing\AppData\Roaming\Typora\typora-user-images\image-20241119220117867.png" referrerpolicy="no-referrer" alt="image-20241119220117867"></p><p><span>	</span><span>计算过程就非常清楚了。那么获得单元应变和单元辅助函数g之后，根据公式11，我们就需要求出它们分别对水平坐标x1和竖直坐标x2的偏导了：</span></p><p><img src="C:\Users\huangyijing\AppData\Roaming\Typora\typora-user-images\image-20241119220446617.png" referrerpolicy="no-referrer" alt="image-20241119220446617"></p><p><span>	</span><span>上图中的公式推导利用的是一种叫高斯积分的方法：对于一个在区域</span><strong><span>[-1, 1]</span></strong><span>上的</span><strong><span>连续且可导</span></strong><span>的函数f(x)来说，有如下等式成立:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n105" cid="n105" mdtype="math_block" data-math-tag-before="0" data-math-labels="[]" data-math-tag-after="0"><div class="md-math-container"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="37.736ex" height="5.835ex" role="img" focusable="false" viewBox="0 -1559 16679.3 2579" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.308ex;"><defs><path id="MJX-640-TEX-LO-222B" d="M114 -798Q132 -824 165 -824H167Q195 -824 223 -764T275 -600T320 -391T362 -164Q365 -143 367 -133Q439 292 523 655T645 1127Q651 1145 655 1157T672 1201T699 1257T733 1306T777 1346T828 1360Q884 1360 912 1325T944 1245Q944 1220 932 1205T909 1186T887 1183Q866 1183 849 1198T832 1239Q832 1287 885 1296L882 1300Q879 1303 874 1307T866 1313Q851 1323 833 1323Q819 1323 807 1311T775 1255T736 1139T689 936T633 628Q574 293 510 -5T410 -437T355 -629Q278 -862 165 -862Q125 -862 92 -831T55 -746Q55 -711 74 -698T112 -685Q133 -685 150 -700T167 -741Q167 -789 114 -798Z"></path><path id="MJX-640-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-640-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-640-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-640-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-640-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-640-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path id="MJX-640-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-640-TEX-N-221A" d="M95 178Q89 178 81 186T72 200T103 230T169 280T207 309Q209 311 212 311H213Q219 311 227 294T281 177Q300 134 312 108L397 -77Q398 -77 501 136T707 565T814 786Q820 800 834 800Q841 800 846 794T853 782V776L620 293L385 -193Q381 -200 366 -200Q357 -200 354 -197Q352 -195 256 15L160 225L144 214Q129 202 113 190T95 178Z"></path><path id="MJX-640-TEX-N-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path id="MJX-640-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-640-TEX-I-1D45C" d="M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z"></path><path id="MJX-640-TEX-N-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msubsup"><g data-mml-node="mo" transform="translate(0 1)"><use data-c="222B" xlink:href="#MJX-640-TEX-LO-222B"></use></g><g data-mml-node="TeXAtom" transform="translate(1046.4,1088.1) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-640-TEX-N-31"></use></g></g><g data-mml-node="TeXAtom" transform="translate(589,-896.4) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-640-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJX-640-TEX-N-31"></use></g></g></g><g data-mml-node="mi" transform="translate(1709.3,0)"><use data-c="1D453" xlink:href="#MJX-640-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(2259.3,0)"><use data-c="28" xlink:href="#MJX-640-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(2648.3,0)"><use data-c="1D465" xlink:href="#MJX-640-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(3220.3,0)"><use data-c="29" xlink:href="#MJX-640-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3887.1,0)"><use data-c="3D" xlink:href="#MJX-640-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(4942.9,0)"><use data-c="1D453" xlink:href="#MJX-640-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(5492.9,0)"><use data-c="28" xlink:href="#MJX-640-TEX-N-28"></use></g><g data-mml-node="mo" transform="translate(5881.9,0)"><use data-c="2212" xlink:href="#MJX-640-TEX-N-2212"></use></g><g data-mml-node="mfrac" transform="translate(6659.9,0)"><g data-mml-node="mn" transform="translate(646.5,676)"><use data-c="31" xlink:href="#MJX-640-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220,-909)"><g transform="translate(853,0)"><g data-mml-node="mn"><use data-c="33" xlink:href="#MJX-640-TEX-N-33"></use></g></g><g data-mml-node="mo" transform="translate(0,89)"><use data-c="221A" xlink:href="#MJX-640-TEX-N-221A"></use></g><rect width="500" height="60" x="853" y="829"></rect></g><rect width="1553" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(8452.9,0)"><use data-c="29" xlink:href="#MJX-640-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(9064.1,0)"><use data-c="2B" xlink:href="#MJX-640-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(10064.3,0)"><use data-c="1D453" xlink:href="#MJX-640-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(10614.3,0)"><use data-c="28" xlink:href="#MJX-640-TEX-N-28"></use></g><g data-mml-node="mfrac" transform="translate(11003.3,0)"><g data-mml-node="mn" transform="translate(646.5,676)"><use data-c="31" xlink:href="#MJX-640-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220,-909)"><g transform="translate(853,0)"><g data-mml-node="mn"><use data-c="33" xlink:href="#MJX-640-TEX-N-33"></use></g></g><g data-mml-node="mo" transform="translate(0,89)"><use data-c="221A" xlink:href="#MJX-640-TEX-N-221A"></use></g><rect width="500" height="60" x="853" y="829"></rect></g><rect width="1553" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(12796.3,0)"><use data-c="29" xlink:href="#MJX-640-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(13407.6,0)"><use data-c="2B" xlink:href="#MJX-640-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(14407.8,0)"><use data-c="1D45C" xlink:href="#MJX-640-TEX-I-1D45C"></use></g><g data-mml-node="mo" transform="translate(14892.8,0)"><use data-c="28" xlink:href="#MJX-640-TEX-N-28"></use></g><g data-mml-node="msup" transform="translate(15281.8,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-640-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,413) scale(0.707)"><use data-c="34" xlink:href="#MJX-640-TEX-N-34"></use></g></g><g data-mml-node="mo" transform="translate(16290.3,0)"><use data-c="29" xlink:href="#MJX-640-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mo>−</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mo stretchy="false">)</mo><mo>+</mo><mi>o</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mn>4</mn></msup><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>	</span><span>其中</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.139ex" height="2.47ex" role="img" focusable="false" viewBox="0 -841.7 2271.6 1091.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-589-TEX-I-1D45C" d="M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z"></path><path id="MJX-589-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-589-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-589-TEX-N-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path><path id="MJX-589-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45C" xlink:href="#MJX-589-TEX-I-1D45C"></use></g><g data-mml-node="mo" transform="translate(485,0)"><use data-c="28" xlink:href="#MJX-589-TEX-N-28"></use></g><g data-mml-node="msup" transform="translate(874,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-589-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)"><use data-c="34" xlink:href="#MJX-589-TEX-N-34"></use></g></g><g data-mml-node="mo" transform="translate(1882.6,0)"><use data-c="29" xlink:href="#MJX-589-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>o</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mn>4</mn></msup><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">o(x^4)</script><span>是一个较小的误差项，可以忽略，此外，通过线性缩放，我们可以将任意区间</span><strong><span>[a, b]</span></strong><span>转换为区域</span><strong><span>[-1, 1]</span></strong><span>, 所以我们有：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n109" cid="n109" mdtype="math_block" data-math-tag-before="0" data-math-labels="[]" data-math-tag-after="0"><div class="md-math-container"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="29.831ex" height="5.835ex" role="img" focusable="false" viewBox="0 -1559 13185.3 2579" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.308ex;"><defs><path id="MJX-641-TEX-LO-222B" d="M114 -798Q132 -824 165 -824H167Q195 -824 223 -764T275 -600T320 -391T362 -164Q365 -143 367 -133Q439 292 523 655T645 1127Q651 1145 655 1157T672 1201T699 1257T733 1306T777 1346T828 1360Q884 1360 912 1325T944 1245Q944 1220 932 1205T909 1186T887 1183Q866 1183 849 1198T832 1239Q832 1287 885 1296L882 1300Q879 1303 874 1307T866 1313Q851 1323 833 1323Q819 1323 807 1311T775 1255T736 1139T689 936T633 628Q574 293 510 -5T410 -437T355 -629Q278 -862 165 -862Q125 -862 92 -831T55 -746Q55 -711 74 -698T112 -685Q133 -685 150 -700T167 -741Q167 -789 114 -798Z"></path><path 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data-mjx-texclass="ORD"><mi>d</mi></mrow></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow><mrow data-mjx-texclass="ORD"><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow><mrow data-mjx-texclass="ORD"><mi>d</mi></mrow></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>y</mi></math></mjx-assistive-mml></mjx-container></div></div><p><span>	</span><span>同样通过线性缩放，我们可以把[a, b], [c, d]区间变化到[-1, 1], [-1, 1]区间，也就是自然坐标系</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.882ex" height="2.262ex" role="img" focusable="false" 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xlink:href="#MJX-700-TEX-N-28"></use></g><g data-mml-node="mo" transform="translate(4753.3,0)"><use data-c="2212" xlink:href="#MJX-700-TEX-N-2212"></use></g><g data-mml-node="mfrac" transform="translate(5531.3,0)"><g data-mml-node="mn" transform="translate(646.5,676)"><use data-c="31" xlink:href="#MJX-700-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220,-909)"><g transform="translate(853,0)"><g data-mml-node="mn"><use data-c="33" xlink:href="#MJX-700-TEX-N-33"></use></g></g><g data-mml-node="mo" transform="translate(0,89)"><use data-c="221A" xlink:href="#MJX-700-TEX-N-221A"></use></g><rect width="500" height="60" x="853" y="829"></rect></g><rect width="1553" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(7324.3,0)"><use data-c="2C" xlink:href="#MJX-700-TEX-N-2C"></use></g><g data-mml-node="mo" transform="translate(7769,0)"><use data-c="2212" xlink:href="#MJX-700-TEX-N-2212"></use></g><g data-mml-node="mfrac" transform="translate(8547,0)"><g data-mml-node="mn" transform="translate(646.5,676)"><use data-c="31" xlink:href="#MJX-700-TEX-N-31"></use></g><g data-mml-node="msqrt" transform="translate(220,-909)"><g transform="translate(853,0)"><g data-mml-node="mn"><use data-c="33" xlink:href="#MJX-700-TEX-N-33"></use></g></g><g data-mml-node="mo" transform="translate(0,89)"><use data-c="221A" xlink:href="#MJX-700-TEX-N-221A"></use></g><rect width="500" height="60" x="853" y="829"></rect></g><rect width="1553" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(10340,0)"><use data-c="29" xlink:href="#MJX-700-TEX-N-29"></use></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing=".5em" columnspacing="1em" displaystyle="true"><mtr><mtd><mo stretchy="false">(</mo><msub><mi>ξ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>η</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mo>,</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><msub><mi>ξ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>η</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mo>,</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><msub><mi>ξ</mi><mn>2</mn></msub><mo>,</mo><msub><mi>η</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mo>,</mo><mo>−</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><msub><mi>ξ</mi><mn>2</mn></msub><mo>,</mo><msub><mi>η</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mo>,</mo><mo>−</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mo stretchy="false">)</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>	</span><span>而根据我们之前的推导，已知自然坐标</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.882ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2157.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-702-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-702-TEX-I-1D709" d="M268 632Q268 704 296 704Q314 704 314 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xlink:href="#MJX-710-TEX-I-1D457"></use></g><g data-mml-node="mi" transform="translate(412,0)"><use data-c="1D44E" xlink:href="#MJX-710-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(941,0)"><use data-c="1D450" xlink:href="#MJX-710-TEX-I-1D450"></use></g><g data-mml-node="mi" transform="translate(1374,0)"><use data-c="1D45C" xlink:href="#MJX-710-TEX-I-1D45C"></use></g><g data-mml-node="mi" transform="translate(1859,0)"><use data-c="1D44F" xlink:href="#MJX-710-TEX-I-1D44F"></use></g><g data-mml-node="mi" transform="translate(2288,0)"><use data-c="1D456" xlink:href="#MJX-710-TEX-I-1D456"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>J</mi><mrow data-mjx-texclass="ORD"><mi>j</mi><mi>a</mi><mi>c</mi><mi>o</mi><mi>b</mi><mi>i</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">J_{jacobi}</script><span>，以及单元在1和2方向上的应变</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.09ex" height="1.462ex" role="img" focusable="false" viewBox="0 -452 2249.8 646" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-754-TEX-I-1D700" d="M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z"></path><path id="MJX-754-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 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data-c="1D700" xlink:href="#MJX-754-TEX-I-1D700"></use></g><g data-mml-node="mn" transform="translate(499,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-754-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(902.6,0)"><use data-c="2C" xlink:href="#MJX-754-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1347.2,0)"><g data-mml-node="mi"><use data-c="1D700" xlink:href="#MJX-754-TEX-I-1D700"></use></g><g data-mml-node="mn" transform="translate(499,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-754-TEX-N-32"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ε</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ε</mi><mn>2</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\varepsilon_1,\varepsilon_2 </script><span>, 以及辅助函数g(r)的值。那么求出对应值之后，带入公式(3)即可计算出样品的J积分:</span></p><p><img src="C:\Users\huangyijing\AppData\Roaming\Typora\typora-user-images\image-20241119221923515.png" referrerpolicy="no-referrer" alt="image-20241119221923515"></p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p></div></div>
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